Answer :
Certainly! Let's solve this problem step-by-step to find the area of sector [tex]\(AOB\)[/tex].
1. Understand the Problem:
- We have a circle with center [tex]\(O\)[/tex].
- The radius [tex]\(OA\)[/tex] (and [tex]\(OB\)[/tex]) of the circle is 5 units.
- The circle's circumference is determined using the formula for the circumference of a circle: [tex]\(C = 2 \pi r\)[/tex].
- We know the length of the arc [tex]\(\widehat{AB}\)[/tex] as a fraction of the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the Total Circumference of the Circle:
- Using the radius [tex]\(OA = 5\)[/tex], the circumference [tex]\(C\)[/tex] of the circle is calculated as follows:
[tex]\[
C = 2 \times \pi \times 5
\][/tex]
- Given [tex]\(\pi = 3.14\)[/tex], substitute this value in:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the Length of Arc [tex]\(\widehat{AB}\)[/tex]:
- The length of the arc [tex]\(\widehat{AB}\)[/tex] is a quarter of the total circumference:
[tex]\[
\text{Length of } \widehat{AB} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
- The area of sector [tex]\(AOB\)[/tex] is a fraction of the total area of the circle, similar to how the arc length is a fraction of the circumference.
- The formula for the area of a sector when the arc length is known is:
[tex]\[
\text{Area of sector } = \frac{\text{Arc length}}{\text{Circumference}} \times \text{Total area of the circle}
\][/tex]
- Total area of the circle is [tex]\(\pi r^2\)[/tex]:
[tex]\[
\text{Total area} = 3.14 \times 5^2 = 78.5 \text{ square units}
\][/tex]
- Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\frac{7.85}{31.4} \times 78.5 = 19.6 \text{ square units}
\][/tex]
So, the area of sector [tex]\(AOB\)[/tex] is closest to 19.6 square units. The correct answer is option A: 19.6 square units.
1. Understand the Problem:
- We have a circle with center [tex]\(O\)[/tex].
- The radius [tex]\(OA\)[/tex] (and [tex]\(OB\)[/tex]) of the circle is 5 units.
- The circle's circumference is determined using the formula for the circumference of a circle: [tex]\(C = 2 \pi r\)[/tex].
- We know the length of the arc [tex]\(\widehat{AB}\)[/tex] as a fraction of the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the Total Circumference of the Circle:
- Using the radius [tex]\(OA = 5\)[/tex], the circumference [tex]\(C\)[/tex] of the circle is calculated as follows:
[tex]\[
C = 2 \times \pi \times 5
\][/tex]
- Given [tex]\(\pi = 3.14\)[/tex], substitute this value in:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the Length of Arc [tex]\(\widehat{AB}\)[/tex]:
- The length of the arc [tex]\(\widehat{AB}\)[/tex] is a quarter of the total circumference:
[tex]\[
\text{Length of } \widehat{AB} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
- The area of sector [tex]\(AOB\)[/tex] is a fraction of the total area of the circle, similar to how the arc length is a fraction of the circumference.
- The formula for the area of a sector when the arc length is known is:
[tex]\[
\text{Area of sector } = \frac{\text{Arc length}}{\text{Circumference}} \times \text{Total area of the circle}
\][/tex]
- Total area of the circle is [tex]\(\pi r^2\)[/tex]:
[tex]\[
\text{Total area} = 3.14 \times 5^2 = 78.5 \text{ square units}
\][/tex]
- Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\frac{7.85}{31.4} \times 78.5 = 19.6 \text{ square units}
\][/tex]
So, the area of sector [tex]\(AOB\)[/tex] is closest to 19.6 square units. The correct answer is option A: 19.6 square units.
